# Focal Definition of Parabola

Parabola is the locus of points equidistant from a given point (focus) and a given line (directrix).

Below is a dynamic illustration of the concept:

The ratio of the two distances is known as eccentricity, it applies to all conic sections; for parabola it is equal to $1.$ Parabola has an axis of symmetry which is perpendicular to the directrix and an vertex - midway between the focus and the directrix. It is customary to think - and this is justified in affine and projective geometries - that, as ellipse and hyperbola, parabola has two foci and two directrices but with one of each located at infinity.

### Equation of Parabola in Cartesian Coordinates

Placing an vertex of the parabola at the origin, the focus at $(0,a)$ and the directrix at $y=-a,$ for some real $a,$ gives, for point $(x,y)$ on the parabola

$y+a = \sqrt{x^{2}+(y-a)^2},$

which, after squaring and simplifications, leads to $4ay=x^2.$ This accounts for $a$ both positive and negative; in the latter case, parabola is turned upside down. For parabola with a horizontal axis, the roles of $x$ and $y$ are reversed: $y^{2}=4ax.$

### Equation of Parabola in Polar Coordinates

In polar coordinates $x=r\mbox{cos}\theta$ and $y=r\mbox{sin}\theta,$ where $r=\sqrt{x^2+y^2}$ is the distance to the origin, and $\theta$ is the angle between the $x-$axis and the vector from the origin to $(x,y).$ It is convenient to shift the parabola down as to place the focus at the origin:

The equations immediately simplifies to $r=r\mbox{sin}\theta+2a,$ or

$\displaystyle r=\frac{2a}{1-\mbox{sin}\theta}.$

For parabola with a horizontal axis the equation becomes $\displaystyle r=\frac{2a}{1-\mbox{cos}\theta}.$

### References

1. D. A. Brannan, M. F. Esplen, J. J. Gray, Geometry, Cambridge University Press, 2002
2. V. Gutenmacher, N. Vasilyev, Lines and Curves: A Practical Geometry Handbook , Birkhauser; 1 edition (July 23, 2004)
3. G. Salmon, Treatise on Conic Sections, Chelsea Pub, 6e, 1960
4. C. Zwikker, The Advanced Geometry of Plane Curves and Their Applications, Dover, 2005
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